Non-over-1 temperament: Difference between revisions

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A ~~'''non-over-2 temperament''' or~~ '''non-over-1 temperament''', is a [[regular temperament]] that tempers a [[subgroup]] corresponding to a harmonic series chord r:n1:...:nk where r ≠ 2, but is not meant to approximate a chord of the form 1:m1:...:mk. Non-over-2 temperaments give regular-temperament interpretations to edos that approximate [[Overtone scales|over-2]] chords such as 4:5:6:7:11 poorly, such as [[14edo]], [[18edo]], [[23edo]] and [[29edo]].

A '''non-over-1 temperament''' is a [[regular temperament]] that tempers a [[subgroup]] corresponding to a harmonic series chord r:n1:...:nk where r ≠ 1, but is not meant to approximate a chord of the form 1:m1:...:mk. Non-over-1 temperaments give regular-temperament interpretations to edos that approximate [[Overtone scales|over-1]] chords such as 4:5:6:7:11 poorly, such as [[14edo]], [[18edo]], [[23edo]] and [[29edo]].

== Examples ==

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Tridec is a temperament generated by a generator near 455.2178¢ (for example, [[29edo|11\29]] or [[37edo|14\37]]) and has an 8-note MOS. If you restrict to the 8-note MOS, Tridec is an 2.7/5.11/5.13/5 temperament that tempers the chord 5:7:11:13; one generator represents a 13/10, three generators represent a 11/10, -4 generators represent a 7/5.

Tridec essentially contains all the notes of 2.3.5 [[porcupine]] temperament and satisfies all its relations; hence it is essentially the same as 13-limit [[Ammonite]]; however, provided you restrict yourself to the 8-note MOS, you're using it as a non-over-2 temperament.

Tridec essentially contains all the notes of 2.3.5 [[porcupine]] temperament and satisfies all its relations; hence it is essentially the same as 13-limit [[Ammonite]]; however, provided you restrict yourself to the 8-note MOS, you're using it as a non-over-1 temperament.

=== Sensi ===

[[Sensi]] is ''effectively'' a non-over-2 temperament provided you restrict yourself to the sensi[8] MOS. The sensi[8] MOS only has a 5:6:7:9:13 chord, but no chord of the form 2:m1:...:mk (except 2:3). Thus sensi can be viewed as a 2.6/5.7/5.9/5.13/10 or 2.3.6/5.7/5.13/10 temperament. (See [http://x31eq.com/cgi-bin/rt.cgi?limit=2_6%2F5_7%2F5_9%2F5_13%2F10&ets=19_27&tuning=po&subgroup=on x31eq data page].)

[[Sensi]] is ''effectively'' a non-over-1 temperament provided you restrict yourself to the sensi[8] MOS. The sensi[8] MOS only has a 5:6:7:9:13 chord, but no chord of the form 2:m1:...:mk (except 2:3). Thus sensi can be viewed as a 2.6/5.7/5.9/5.13/10 or 2.3.6/5.7/5.13/10 temperament. (See [http://x31eq.com/cgi-bin/rt.cgi?limit=2_6%2F5_7%2F5_9%2F5_13%2F10&ets=19_27&tuning=po&subgroup=on x31eq data page].)

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-A '''non-over-2 temperament''' or '''non-over-1 temperament''', is a [[regular temperament]] that tempers a [[subgroup]] corresponding to a harmonic series chord r:n<sub>1</sub>:...:n<sub>k</sub> where r ≠ 2, but is not meant to approximate a chord of the form 1:m<sub>1</sub>:...:m<sub>k</sub>. Non-over-2 temperaments give regular-temperament interpretations to edos that approximate [[Overtone scales|over-2]] chords such as 4:5:6:7:11 poorly, such as [[14edo]], [[18edo]], [[23edo]] and [[29edo]].
+A '''non-over-1 temperament''' is a [[regular temperament]] that tempers a [[subgroup]] corresponding to a harmonic series chord r:n<sub>1</sub>:...:n<sub>k</sub> where r ≠ 1, but is not meant to approximate a chord of the form 1:m<sub>1</sub>:...:m<sub>k</sub>. Non-over-1 temperaments give regular-temperament interpretations to edos that approximate [[Overtone scales|over-1]] chords such as 4:5:6:7:11 poorly, such as [[14edo]], [[18edo]], [[23edo]] and [[29edo]].
 == Examples ==
@@ Line 5: / Line 5: @@
 Tridec is a temperament generated by a generator near 455.2178¢ (for example, [[29edo|11\29]] or [[37edo|14\37]]) and has an 8-note MOS. If you restrict to the 8-note MOS, Tridec is an 2.7/5.11/5.13/5 temperament that tempers the chord 5:7:11:13; one generator represents a 13/10, three generators represent a 11/10, -4 generators represent a 7/5.
-Tridec essentially contains all the notes of 2.3.5 [[porcupine]] temperament and satisfies all its relations; hence it is essentially the same as 13-limit [[Ammonite]]; however, provided you restrict yourself to the 8-note MOS, you're using it as a non-over-2 temperament.
+Tridec essentially contains all the notes of 2.3.5 [[porcupine]] temperament and satisfies all its relations; hence it is essentially the same as 13-limit [[Ammonite]]; however, provided you restrict yourself to the 8-note MOS, you're using it as a non-over-1 temperament.
 === Sensi ===
-[[Sensi]] is ''effectively'' a non-over-2 temperament provided you restrict yourself to the sensi[8] MOS. The sensi[8] MOS only has a 5:6:7:9:13 chord, but no chord of the form 2:m<sub>1</sub>:...:m<sub>k</sub> (except 2:3). Thus sensi can be viewed as a 2.6/5.7/5.9/5.13/10 or 2.3.6/5.7/5.13/10 temperament. (See [http://x31eq.com/cgi-bin/rt.cgi?limit=2_6%2F5_7%2F5_9%2F5_13%2F10&ets=19_27&tuning=po&subgroup=on x31eq data page].)
+[[Sensi]] is ''effectively'' a non-over-1 temperament provided you restrict yourself to the sensi[8] MOS. The sensi[8] MOS only has a 5:6:7:9:13 chord, but no chord of the form 2:m<sub>1</sub>:...:m<sub>k</sub> (except 2:3). Thus sensi can be viewed as a 2.6/5.7/5.9/5.13/10 or 2.3.6/5.7/5.13/10 temperament. (See [http://x31eq.com/cgi-bin/rt.cgi?limit=2_6%2F5_7%2F5_9%2F5_13%2F10&ets=19_27&tuning=po&subgroup=on x31eq data page].)
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